It has long been known to perform signal analysis using a sinusoidal analyzing signal. Well-known techniques of this type include Fourier analysis and the discrete cosine transform. Typical applications of these techniques include spectral analysis and data compression.
Signal analysis using a sinusoidal analyzing signal is well-suited for analysis of so-called "stationary" signals, i.e., signals whose characteristics remain unchanged over relatively long periods of time. However, there are difficulties in attempting to analyze signals exhibiting non-stationary characteristics on the basis of sinusoidal signals. In attempting to overcome these difficulties, it has been proposed to perform the signal analysis over limited time "windows" or, similarly, to use an analyzing signal constructed by multiplying a sinusoidal function with a windowing function. Such "windowed" analyzing functions have been applied with some degree of success to analysis of signals exhibiting short-term behavior but there has remained a need to improve the unavoidable trade-off of frequency resolution against time resolution.
Another type of signal analysis known as the wavelet transform has recently been proposed.
The basis functions used in the wavelet,transform are obtained from a single prototype or basis wavelet ("mother wavelet") h(t) by applying a scaling factor and a translation factor to provide a "family" of wavelets: ##EQU1##
The scaling factor a stretches or shrinks (dilates or compresses) the basis wavelet, while the translation factor b shifts the basis wavelet in time. For discrete analysis it has been found convenient to limit the scaling factor a to powers of two, i.e., a=2.sup.m, (m being an integer) and to limit the translation factor b to integers n, it being understood that m and n may be negative or zero as well as positive. It has also been proposed to perform a wavelet transform on a discrete sequence of signals by a filter bank algorithm as schematically illustrated in FIG. 1.
In FIG. 1 a signal sequence x(n) is input into a multi-stage filter bank formed of a cascaded halfband lowpass blocks LP1, LP2, LP3, etc.; halfband highpass blocks HP1, HP2, HP3, etc.; and decimation blocks D1, D2, D3, D4, D5, D6, etc. Each of the halfband lowpass blocks is identical and is implemented using a scaling function .phi.(x). The cascaded lowpass blocks provide a multiresolution analysis of the input signal, and each halfband highpass block is identical, except for time reversal, to the standard lowpass blocks LP1, LP2 etc. so that each signal band can be perfectly reconstructed from the outputs of the respective lowpass and highpass blocks for the band. The decimation blocks carry out decimation by a factor of two; that is, every other sample is discarded. The desired wavelet transform coefficients in the respective signal bands (octaves) are provided as the output signals from the decimation blocks D1, D3, D5, etc. Wavelet filter banks have been described, for example, in "Wavelets and Filter Banks: Theory and Design," M. Vetterli and C. Herley, IEEE Transactions on Signal Processing, 1992; "A Theory for Multiresolution Signal Decomposition: The Wavelet Representation," S. Mallat, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 11, no. 7, July 1989, pp. 674-693; "Wavelets and Filter Banks: Relationships and New Results," M. Vetterli and C. Herley, Proceedings, 1990 IEEE International Conference on Acoustics, Speech and Signal Processing, April 1990, pp. 1723-1726; "Wavelets and Signal Processing," O. Rioul and M. Vetterli; IEEE SP Magazine, October 1991, pp. 14-38.
It has been proposed to employ wavelet transform analysis in data compression, image analysis, computer vision, speech recognition and seismic signal analysis, among other applications. It has also been suggested that wavelet transform analysis could be applied to detection of transient signals. ("Use of the Wavelet Transform for Signal Detection," N. Erdol and Filiz Basbug, Proceedings of SPIE 1993 International Symposium on Optical Engineering and Photonics in Aerospace and Remote Sensing, April 1993, vol. 1961, pp. 401-410; "Detection of Transient Signals by the Gabor Representation," B. Friedlander and B. Porat, IEEE Trans. Acoust., Speech, Signal Proc., vol. 38, 1992, pp. 169-180; "Wavelet Transformations in Signal Detection," F. B. Tuteur, Proceedings of ICASSP-88, 1988, pp. 1435-38 (detection of abnormalities in electrocardiograms)). However, prior approaches to wavelet-based transient detection have not dealt satisfactorily with the sensitivity of the wavelet transform to the phase of the input signal. Because wavelet transform analyses suitable for detecting the transient signals generally lack phase-invariance, shifts in the initial phase of the target signal result in significant "leakage" in wavelet transform coefficients between translations. Where the time of occurrence of the target signal is not known in advance, the leakage between translations can prevent signal detection. It has been proposed to perform transient signal detection by matching against a translation--invariant template wavelet coefficients that differ substantially from zero ("The Use of the Wavelet Transform in the Detection of an Unknown Transient Signal," M. Frisch and N. Messer, IEEE Transactions on Information Theory, vol. 38, No. 2, March 1992, pp. 892-897) but, such an approach has not been found to perform very successfully, and also entails computational complexity.
It also has not heretofore been recognized that wavelet transform analysis can be applied to electronic article surveillance (EAS) systems, and particularly to detection of a signal generated by an EAS marker in response to an interrogation signal field formed by an EAS system. Typically, the EAS marker signal is received along with correlated noise signals of higher amplitude than the marker signal. Also, the amplitude and the time of occurrence of the marker signal are not known in advance. Reliable detection of the marker signal consequently presents significant challenges.